Secondary II - PBworksnucc.pbworks.com/w/file/fetch/62335003/Unit_01_Similarity... · Web view1. 15...

Unit 1

SimilarityStudent Edition

Northern Utah Curriculum Consortium

Project LeaderSheri Heiter

Weber School District

Project Contributors

Ashley MartinDavis School District

Bonita RichinsCache School District

Craig AshtonCache School District

Gerald JackmanBox Elder School District

Jeff RawlinsBox Elder School District

Jeremy YoungBox Elder School District

Kip MottaRich School District

Marie FitzgeraldCache School District

Mike HansenCache School District

Robert HogganCache School District

Sheena KnightWeber School District

Teresa BillingsWeber School District

Wendy BarneyWeber School District

Helen HeinerDavis School District

Susan SummerkornDavis School District

Lead EditorAllen JacobsonDavis School District

Technical Writer/EditorDianne CumminsDavis School District

NUCC |Secondary II Math i

Table of Contents1.1 SIMILAR FIGURES...............................................................................................................................4

Student Notes.............................................................................................................................................4

Similarity 1 – How Are We Related? A Develop Understanding Task 1a................................................5

Similarity 1 – Discovering Similarity A Develop Understanding Task 1b...............................................7

Ready, Set, Go!..........................................................................................................................................9

1.2 A SIMILAR TRIANGLES....................................................................................................................11

Student Notes...........................................................................................................................................11

A Similar Triangles Conjecture A Develop Understanding Task 2.........................................................12

Ready, Set, Go!........................................................................................................................................13

1.3 POSTULATES FOR SIMILAR TRIANGLES.....................................................................................15

Student Notes...........................................................................................................................................15

Similar Triangles Postulates A Solidify Understanding Task 3...............................................................16

Ready, Set, Go!........................................................................................................................................17

1.4 CONGRUENCE AND SIMILARITY..................................................................................................19

Student Notes...........................................................................................................................................19

Similarity 4 – T.V.’s and Shadows A Practice Understanding Task 4...................................................20

Ready, Set, Go!........................................................................................................................................21

1.5 PROVING THEOREMS.......................................................................................................................24

Student Notes...........................................................................................................................................24

Cool Things with Triangles A Develop Understanding Task 5...............................................................25

Ready, Set, Go!........................................................................................................................................27

1.6 PROVING RELATIONSHIPS IN GEOMETRIC FIGURES...............................................................30

Student Notes...........................................................................................................................................30

We Are Similar, Can You Tell? A Solidify Understanding Task 6.........................................................31

1.7 VERIFYING DILATIONS 1.................................................................................................................33

Student Notes...........................................................................................................................................33

1.7 VERIFYING DILATIONS 2.................................................................................................................34

Student Notes...........................................................................................................................................34

Double the Logo A Develop Understanding Task 7a..............................................................................35

Dilation and Similarity A Develop Understanding Task 7b...................................................................36

Ready, Set, Go!........................................................................................................................................37

1.8 SIMILAR FIGURES.............................................................................................................................40

Student Notes...........................................................................................................................................40

NUCC |Secondary II Math ii

Find the Center A Solidify Understanding Task 8..................................................................................41

Ready, Set, Go!........................................................................................................................................42

1.9 PARTITIONING A LINE SEGMENT.................................................................................................45

Student Notes...........................................................................................................................................45

The Divided Line A Solidifying Understanding Task 9...........................................................................46

Ready, Set, Go!........................................................................................................................................48

HOMEWORK HELP FOR STUDENTS AND PARENTS........................................................................50

SIMILARITY ASSESSMENT....................................................................................................................51

NUCC |Secondary II Math iii

Unit 1.1

1.1 SIMILAR FIGURESStudent Notes

NUCC | Secondary II Math 4

Unit 1.1

Similarity 1 – How Are We Related?A Develop Understanding Task 1a

Name_____________________ Hour___________

Kyle was working on a problem his teacher gave him. He was supposed to divide several pairs of figures into two categories and this is what he has come up with:

Group 1: Group 2:

Explain what reasons Kyle may have had for this grouping. Be as specific as possible.


Unit 1.1


Unit 1.1

Based on your explanation decide into which group would Tran put these three pairs?

Group ______, because




FE

H G

CD

BA

X

W

A B

Unit 1.1

Similarity 1 – Discovering SimilarityA Develop Understanding Task 1b

Name_________________________________________ Hour___________

Take a minute to recall what you know about congruent figures.

Congruence:

Figure 1These two shapes ARE Similar. Using your centimeter ruler and protractor, measure all of the sides and angles. Label the measures of the sides and the measures of the angles on the picture.

1. What do you notice about the measurements of the corresponding angles?

2. What do you notice about the measurements of the corresponding sides?

3. When you compare AB∧side EF , are they congruent?

4. Can you find any relationship between side AB∧side EF ?

5. Does this relationship hold (is it true) for all other corresponding sides?

Figure 2


Unit 1.1

The following two shapes are similar. Measure the following sides and angles. Label these measurements on your picture.

1.How are the shapes in Figure 1 different from the shapes here in Figure 2?

This means that other shapes can be similar as well as long as certain rules are true.

2. What do you notice about the measurements of the corresponding angles?

3. What do you notice about the measurements of corresponding sides?

4. When you compare side AB∧side XY are they congruent?

5. Explain the relationship between side AB∧side XY .

6. Does this relationship hold (is it true) for all other corresponding sides? (Measure)

Figure 3These two figures are NOT similar figures.

7.Explain WHY the two trianglesabove are not similar.

8. Summary – What makes two figures similar? (Write a general rule you have discovered to be true.)


B

A

C

D

GF

HE

A A

A A

A

A

A

A

Unit 1.1

Ready, Set, Go!Name_______________________________ Hour____________

Ready

Solve each proportion for the given variable.

1. 15y

=4012 2. 16

40=30

x 3.y

42.3= 144

56.4

4. 126k

=143 5. 2x+1

2= x+2

5 6. 25= x

21−x

Set

Is each given similarity statement true or false? Take measurements to decide if needed. EXPLAIN your answer.

7. ABCD EFGH

8. ABCD EFGH


4

8

6

BC

A

F

EED

43

2

50◦130◦ 50◦

130◦1 1

3

3

6

6

2 2

140◦40◦

140◦ 40◦

Unit 1.1

State whether each of the following illustrates two similar figures. How do you know?

9. 10.

Go!

Assume that ∆ PLU ∆ ABC . Find AC∧BC (label∈answer ) for eachof the givenlengthsof AB.

Math Composer 1.1. 5ht tp: / / www.mathcomposer.com

C

BA4

56

U

LP (not to scale)

11. AB=1 12. AB=3

13. AB=4 14. AB=x

∆ CAT ∆ .̇Complete eachstatement .

15. ∠C≅____________ 16. ∠CTA ≅ ___________ 17. ∠DTO≅ __________18. ∠A≅_____________19. ∠ D≅ __________ 20. ∠O≅_____________


Unit 1.2

1.2 A SIMILAR TRIANGLESStudent Notes


B

C

A

E

F

D

320

320

400

400

Unit 1.2

A Similar Triangles ConjectureA Develop Understanding Task 2

Name_________________________________________ Hour___________

In the triangles to the right, ∠ A≅∠D∧∠B≅∠E .

What is the measure of ∠C and why?

What is the measure of ∠F and why?

Measure the sides and compare the ratios of corresponding side lengths.

Is ABDE

≈ ACDF

≈ BCEF ?

Is ∆ ABC ∆≝¿ ? Why or why not?

Now, draw your own ∆ ABC with different side lengths between 2cm and 5cm long.

Construct a 2nd triangle, ∆ XYZ, above as well, but with different side lengths of ∆ ABC. Make this triangle not congruent to ∆ ABC but with ∠X=∠A and ∠Y=∠B . (use a protractor and ruler to help you)Are your two triangles ∆ ABC∧∆ XYZsimilar? Explain.


Q

10 ft

600

P

A

C

B600 2.5 ft

Unit 1.2

Ready, Set, Go!Name_______________________________ Hour____________Ready

1. How can you determine if two polygons are similar?

Explain if each statement is true or false. Explain your reasoning. 2. All squares are similar.

3. If an acute angle of a right triangle is congruent to an acute angle of a second right triangle, then the triangles are similar.

4. All isosceles triangles are similar.

For #5-7, ∆ ACB ∆ PCQ . (See figure on right5. Find m∠Q .

6. Find AQ.

7. Find AB.

Set

For problems #8-11, ∆ ABC ∆ XYZ . For each problem, sketch and label the given angles or sides for ∆ ABC∧∆ XYZ . Then solve for t.

8. m∠ A=100 ° m∠B=60 ° m∠Z=t

9. AB=8 cm BC=10 cm XY=12cmYZ=t

10. AB=t AC=15 ft XY =9 ft XZ=6 ft

11. CB=14 mCA=5 mZX=7m ZY=t +1


k

ms

trn\n\\\

B C5

x x-4

EA

D

F3

Unit 1.2

12. Alicia and Jason were writing proportions for similar triangles shown at the right.

Who is correct? Explain your reasoning.

13. Identify the similar triangles (write a similarity statement). Find x and the measures of the sides AB and DE.

Go!

15. To make concrete for the footings of her new deck, Marie mixes cement and sand in a ratio of 2 bags cement to 3 bags of sand. How many bags of sand will she need if she uses 32 bags of cement?

16. A 212-pound person weighs about 34 pounds on the moon. About how much will a 130-pound person weight on the moon?

17. A baker’s recipe calls for 6 cups of flour for every 2 cups of milk. If the baker uses 9 cups of flour, how many cups of milk will he need?

18. A machine worker can produce 75 auto parts in 1.5 hours. How many auto parts can the worker produce in 5 hours?

19. A survey shows that four out of every seven students watches TV more than three hours per day. If there are 560 seniors in a high school, how many would you expect to watch more than three hours of TV every day?

20. The advertisement for a scale model of a classic automobile gives the scale of the model

as 1 to 24. The model is 714 inches long. What is the actual length of the automobile?


Alicia

rk= s

m

rm=ks

Jason

rk=m

s

rs=km

Unit 1.3

1.3 POSTULATES FOR SIMILAR TRIANGLESStudent Notes


Unit 1.3

Similar Triangles PostulatesA Solidify Understanding Task 3

Name_____________________ Hour___________

In the last lesson we learned the AA (Angle-Angle) postulate. Today we are going to extend our “similarity tool box” to include other postulates.

Directions: - In the box labeled “A” use a ruler to draw only two sides of a triangle with their

measurements between 2 cm and 5 cm. Label these measurements. - In the box labeled “B” use a ruler to draw only two sides of a triangle with measurements

proportional to those drawn in box “A.” (Hint: increase the measurements by the same amount.) Label these measurements.

- Next, connect your 3rd side to create the triangles. Measure these lengths and label them. Are your 3rd sides of the triangles proportional to each other?

A B

Follow up Questions:When you know 2 sides of a triangle are proportional (you drew your example this way), will your 3rd side always be proportional also?

Turn to a neighbor and compare. Use an additional paper to experiment if needed.

Are your two triangles in box “A” and “B” similar? (Borrow a protractor to measure the angles)


Unit 1.3

Ready, Set, Go!Name_________________________________________________________Hour____________

Ready

1. How does the SSS postulate for similar triangles differ from the SSS postulate for congruent triangles?

2. How does the SAS postulate for similar triangles differ from the SAS postulate for congruent triangles.

Set

List the similar triangles in the following figures. Write the postulate or theorems that prove your answer.

3. 4.

Math Composer 1. 1. 5http: / / www. mathcomposer. com

70º

110º

YX

C

B A

Math Composer 1. 1. 5http: / / www.mathcomposer. com

12 ft8 ft

4 ft6 ft

P

DC

BA


Unit 1.3

Use the diagram to complete Exercises 5-8.

Math Composer 1. 1. 5ht tp: / /www.mathcomposer. com

R

Q

M

L

P

5. PLPQ

= PM? 6.

?LQ

= PMMR 7.

PL+LQLQ

= ?MR

8. If LQ=9cm, PM=8cm ,∧MR=12 cm, then PL=¿ ___________cm.

Go!

Determine whether each pair of triangles is similar. Justify your answer.

9. 10. Math Composer 1. 1. 5ht tp: / / www. mathcomposer .com

FE

D

C B

A

108

4 5

Math Composer 1. 1. 5http: / / www. mathcomposer.com

P

D

C

B

A

11.Math Composer 1.1.5http: / /www.mathcomposer .com

3

7F

E

D

C

B

A

25

21

925/3


Unit 1.4

1.4 CONGRUENCE AND SIMILARITYStudent Notes


Unit 1.4

Similarity 4 – T.V.’s and ShadowsA Practice Understanding Task 4

Name___________________________________________ Hour___________

1. Most TV screens have similar shapes. The measure of the diagonal is used to give screen size. Suppose the dimensions of a 9-inch screen are 5.5 inches by 7.5 inches. Find the dimensions of an 18-inch TV and a 35-inch TV.

2. A 9ft tall stop sign casts a 12ft shadow. A building near this stop sign casts a 63ft shadow.a. How tall is the building?b. If the distance from the top of the building to the end of the shadow is 87ft, what is the

distance from the top of the stop sign to the end of its shadow?


Unit 1.4

Ready, Set, Go!Name___________________________________________________________ Hour____________Ready

1. Find the length of the lake.

2. A Tower casts a shadow of 64 feet. A 6-foot pole near the tower casts a shadow 8 feet long. How tall is the tower?

3. A ladder that is 30ft tall leans 25ft up against the side of a building. Up against the same building, how far up would a 20ft ladder go?

4. The triangles in the figure are similar. Find the value of x.

5. Maria is visiting the Washington Monument in D.C. She wants to know the height of the Monument. The monument’s shadow is 111 feet at the same time that Maria’s shadow is 1 foot Maria is 5 feet tall.

Set

6. Sam built a ramp to a loading dock. The ramp has a vertical support 2 meters from the base of the loading dock and 3 meters from the base of the ramp. If the vertical support is 1.2 meters in height, what is the height of the loading dock?


Math Composer 1. 1. 5http: / / www. mathcomposer. com

24 cm

62.4 cmx13cm

5 cm

12 cm

h 6’

24’48’

Unit 1.4

7. Two extension ladders are leaning at the same angle against a vertical wall. The 3-m ladder reaches 2.4 m up the wall. How far up the wall does the 8-m ladder reach?

8. Mr. Smith is having some photos enlarged for his home. He wants to enlarge a photo that is 5 inches by 7 inches so the dimensions are 3 times larger than the original. How many times larger than the original photo will the area of the new photo be?

9. Emily is moving and needs to pack two mirrors. The largest mirror fits in a box that is 18 inches by 20 inches long. Her smaller mirror is similar in proportion to the larger mirror. Emily determines that the width of the smaller box needs to be a minimum of 9 inches. What should the minimum length of the box be to hold the smaller mirror?

Go!

For # 10 and 11 A) find the measurement and B) explain how you know the triangles are similar. Circle your answers.

10. A flagpole casts a shadow 48 feet long at the same time that a 6 foot tall person casts a shadow 24 feet long. How tall is the flagpole?

A)

B)


Lake

A

B

50 m

50 m

P

25 m

D

15 m

C25 m

Unit 1.4

11. What is the distance of AB across the lake?

A)

B)


Unit 1.5

1.5 PROVING THEOREMSStudent Notes


Unit 1.5

Cool Things with TrianglesA Develop Understanding Task 5

Name__________________________________________ Hour___________1. Using a ruler, draw any kind of triangle. Now, draw a line through the triangle that is parallel

to one side. This creates two triangles, the original and a smaller one. What so you notice about the two triangles?Compare this with your partner(s) picture, is the same thing true?

2. The two triangles in the picture below are proportionate. What is true about sides BC and DE? How do you know?

3. If the three sides of ∆UVW are proportionate to the corresponding sides of ∆KLM, are the triangles similar? How do you know?


http://www.google.com/imgres?q=similar+triangles&hl=en&sa=X&biw=1920&bih=940&tbm=isch&prmd=imvns&tbnid=3066nrcq71E8qM:&imgrefurl=http://library.thinkquest.org/20991/geo/spoly.html&docid=qkK6WunSdWuNoM&imgurl=http://library.thinkquest.org/20991/media/geo_spsas.gif&w=300&h=300&ei=lXQUUMfRFYXFqgH1_IHwDA&zoom=1&iact=hc&vpx=1480&vpy=2&dur=3555&hovh=225&hovw=225&tx=96&ty=78&sig=101416040128969081115&page=1&tbnh=138&tbnw=138&start=0&ndsp=48&ved=1t:429,r:46,s:0,i:285

Unit 1.5

4. Using similar triangles to prove the Pythagorean Theorem.

a. There are three separate triangles in this picture. Draw the 3 triangles below, being careful to draw them in the same orientation, and include their vertices (A-D) and side lengths (a-f).

b. Write the similarity statements of the three triangles (∆XYZ ∆QRS)

c. Write the proportions showing the short side and the hypotenuse for the small and large triangles. Then write the proportions showing the long sides and the hypotenuse for the medium and large triangles.

d. Take those two proportion statements and cross multiply, resulting in two equations.

e. Add those two equations together.

f. Show how you can simplify this equation to get the Pythagorean Theorem.


Unit 1.5

Ready, Set, Go! (with some solutions)

Name_____________________________ Hour____________Ready

Are the following triangles similar? If so, write a similarity statement.

1. 2.

3. 4.

5. 6.

7. 8.


Unit 1.5

Set

In order to estimate the width of a river, the following technique can be used. Use the diagram on the left.

Place three markers, and on the upper bank of the river. is on the edge of the river and . Go across the river and place a marker, so that it is collinear with and . Then,

walk along the lower bank of the river and place marker , so that . feet, feet, feet.

1. Is How do you know?

2. Is How do you know?

3. What is the width of the river? Find .

4. Can we find ? If so, find it. If not, explain.

5. The technique above was used to measure the distance across the Grand Canyon. Using the same set up and marker letters, , and . Find

(the distance across the Grand Canyon).

Go

6. Cameron is 5 ft tall and casts a 12 ft shadow. At the same time of day, a nearby building casts a 78 ft shadow. How tall is the building?

7. The Empire State Building is 1250 ft. tall. At 3:00, Pablo stands next to the building and has an 8 ft. shadow. If he is 6 ft tall, how long is the Empire State Building’s shadow at 3:00?

8.


Unit 1.5

8. A tree outside Ellie’s building casts a 125 foot shadow. At the same time of day, Ellie casts a 5.5 foot shadow. If Ellie is 4 feet 10 inches tall, how tall is the tree?

Solution: Draw a picture. We see that the tree and Ellie are parallel, so the two triangles are similar.

The measurements need to be in the same units. Change everything into inches and then we can cross multiply.

Answer questions 17-20 about trapezoid .

9. Name two similar triangles. How do you know they are similar?

10. Write a true proportion.

11. Name two other triangles that might not be similar.

12. If and , find . Be careful!


Unit 1.6

1.6 PROVING RELATIONSHIPS IN GEOMETRIC FIGURESStudent Notes


Unit 1.6

We Are Similar, Can You Tell?A Solidify Understanding Task 6

Name_____________________ Hour___________

The general formula for a circle is (x – h)2 + (y – k)2 = r2, with center (h, k) and the radius r.1. Choose three numbers between -5 and 5: a = _____ b = _____ c = _____

Graph the circle with center (a, b) and radius |c|.Write your neighbor's numbers here: a = _____ b = _____ c = _____ and graph their circle.Record your circle's equation: ______________________________Record your neighbor's circle's equation:__________________________________Show that the two circles are similar.


Unit 1.6

The equation of a parabola in vertex form is y = a(x – h)2 + k, with vertex (h, k) and lead coefficient a.

2. On the grid below graph the parabola with your numbers a as the lead coefficient, and vertex (b, c), as well as the parabola your neighbor drew. You may need to make a table or use a graphing calculator.

Record your equation: __________________________________

Record your neighbor's equation: __________________________

Show that the two parabolas are similar.


Unit 1.7

1.7 VERIFYING DILATIONS 1Student Notes


Unit 1.7

1.7 VERIFYING DILATIONS 2Student Notes


Unit 1.7

Double the LogoA Develop Understanding Task 7a

Name_________________________________________ Hour___________Your assignment is to double the logo drawn below.

Take two rubber bands of equal length and tie them together so that there are equally sized bands on either side of the knot. Pick and mark an anchor point somewhere on the paper. Pin the end of rubber band to the anchor point with your finger. On the opposite side of the other band, place a pen. Trace a new object while keeping the knot consistently on top of the figure you are trying to enlarge.

Your task is to:1. Follow directions as accurately as you can, but don’t worry if your pictures are

wiggly.2. Explain what in this procedure caused the shape to be twice the size of the original

one. What happens if you choose a different anchor point?3. How can you perform the same process if instead of rubber bands you used a ruler?4. List as many different things you notice that

a. stayed the sameb. changed in this process

Depending on where you put your anchor point, your new picture might want to run off the page. If this is the case, tape another sheet of paper where it wants to be.


Unit 1.7

Dilation and SimilarityA Develop Understanding Task 7b

Name___________________________________________ Hour___________

Draw an axis and copy the shape to the left on your graph paper.

Have each group member choose a different number between 2 and 6. Use the slope triangles as shown to slide each point A, B, C, and D along the line connecting the point to the origin. If your number is 3, use 3 slope triangles, as drawn in the picture. If your number is 4, use 4 triangles. If your number is a, draw a triangles.

Once you have relocated points A, B, C, D, connect them with segments to form a new quadrilateral. Cut out the new quadrilateral.

1. How are the quadrilaterals in your group alike?

2. How are they different?

3. What is the relationship between your quadrilateral and the other quadrilaterals in your group?

4. How do you know?

5. If you move the point of dilation 4 units to the right, is the relationship you found earlier maintained? How would you describe this relationship?

6. Will this relationship be true of any polygon place anywhere on the coordinate grid? How do you know?


Unit 1.7

Ready, Set, Go!Name______________________________________________________________ Hour____________Ready

Practice with Proportions: Solve each equation below for x. Show all work and check your answer by substituting it back into the equation and verifying that it makes the equation true.

1. x3=62. 5 x+9

2=12

3. x4=9

64. 5

x=20

8

Practice with dilations:

5. If a segment AB having length 2 in. is dilated with a scale factor of 3, what is the length of A’B’?

6. If an equilateral triangle ∆TRI whose side length is 4 m. is dilated with a scale factor of 1/2, what is the length of the sides of T’R’I’?

7. If a square SQRE that had side lengths of 5cm has been dilated to S’Q’R’E’ with side lengths of 15cm, what was the scale factor of the dilation?

Set

8. Plot the rectangle ABCD formed with the points A(−1, −2), B(3, −2), C(3, 1), and D(−1, 1) onto the graph. Use the method from the problem task to enlarge it from the origin by a factor of 2 (using two “rubber bands” or slope triangles). Label this new rectangle A′B′C′D′.

a. What are the dimensions of the enlarged rectangle, A′B′C′D′?b. Find the area and the perimeter of A′B′C′D ′.c. Find AC (the length of AC).


Unit 1.7

9. The center of dilation is Pand the scale factor is 3. Find PQ' .

Solution: If the scale factor is 3 and Qis 6 units away from P, then Q'is going to be 6 ∙ 3=18units away from P. The dilation will be on the same line as the original and center.

10. Using the picture above, change the scale factor to 13 . Find Q' ' .

Solution: The scale factor is 13 , so Q ' 'is going to be 6 ∙ 1

3=2units away from P. Q ' 'will

also be collinear with Qand center.11. KLMN is a rectangle. If the center of dilation is Kand k=2, draw K ' L' M ' N '.

Solution: If K is the center of dilation, then Kand K 'will be the same point. From there, L'will be 8 units above Land N ' will be 12 units to the right of N.

12.Find the perimeters of KLMNand K ' L' M ' N '. Compare this ratio to the scale factor.

Solution: The perimeter of KLMN=12+8+12+8=40. The perimeter of K ' L' M ' N '=24+16+24+16=80. The ratio is 80:40, which reduces to 2:1, which is the same as the scale factor.


Unit 1.7

13. ABCis a dilation of ∆≝¿. If Pis the center of dilation, what is the scale factor?

Solution: Because ∆ ABC is a dilation of ∆≝¿, then ∆ ABC ∆≝¿. The scale factor is the ratio of the sides. Since ∆ ABC is smaller than the

original, ∆≝¿, the scale factor is going to be less than one, 1220

=35 .

Go!

For the following problems, plot the points, connect the points with lines to create the shape, find the perimeter and area, then dilate it with the origin as the center of dilation by the given scale factor.

14.A(-6, 0) B(4, 0) C(4, 8) D(-6, 8) ; k = ½ 15. X(-5, -2) Y(-5, 2) Z(-1,2) ; k=3

P = P =

A = A =


Unit 1.7


Unit 1.8

1.8 SIMILAR FIGURESStudent Notes


Unit 1.8

Find the CenterA Solidify Understanding Task 8

Name_____________________ Hour___________

1. Rectangle A′B′C′D′ is the image of rectangle ABCD under a certain dilation. Find the center of dilation P and the scale factor. Explain how you found it and why your method works.

2. The small rectangle A′B′C′D′ is the image of rectangle ABCD under a certain dilation. Find

the center of dilation P. What can you say about the scale factor?

3. Triangle A′B′C′ has the same shape as triangle ABC, but can’t possibly be the image of

triangle ABC under any dilation. Explain why not.

4. Determine a sequence of transformations that will take the triangle ABC to triangle A′B′C′.


Unit 1.8

Ready, Set, Go!Name_______________________________________________________ Hour____________Ready

Given Aand the scale factor k , determine the coordinates of the dilated point, A '. You may assume the center of dilation is the origin.

1. A (3,9 ) , k=23 2. A (−4,6 ) , k=2

Given Aand A ', find the scale factor. You may assume the center of dilation is the origin.

3. A (8,2 ) , A ' (12,3 ) 4. A (22 ,−7 ) , A '(11 ,−3.5)

For the given shapes, draw the dilation, given the scale factor and center.

5. k=2

, center is A

6.k=3

4,

center is A

In the two questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the black figure is the original and Pis the center of dilation.

7. k=4 8. k=13


Unit 1.8

In the two questions below, find the scale factor, given the corresponding sides. In each diagram, the black figure is the original and is the center of dilation.

9. 10.

Set In this text, the center of dilation will always be the origin.

11. Quadrilateral EFGHhas vertices E (−4 ,−2 ) , F (1,4 )G (6,2 )∧H (0 ,−4 ). Draw the dilation with a scale factor of 1.5.

Solution: To dilate something in the coordinate plane, multiply each coordinate by the scale factor. This is called mapping.For any dilation the mapping will be (x , y )→ (kx , ky).For this dilation, the mapping will be (x , y )→ (1.5 x ,1.5 y ). In the graph above, the blue quadrilateral (inside) is the original and the red image(outside) is the dilation.

12. Determine the coordinates of ∆ ABC and ∆ A ' B ' C ' and find the scale factor.

Solution: The coordinates of ∆ ABC are (2 , 1 ) , B (5 , 1 )∧C (3 , 6). The coordinates of ∆ A ' B ' C ' are A' (6 ,3 ) ,B' (15 , 3 )∧C '(9 ,18) . Each of the corresponding coordinates are three times the original, so k=3.

13. Show that dilations preserve shape by using the distance formula. Find the lengths of the sides of both triangles in Example 12.

Solution: d ( AB )=3 , d ( AC )=√ (3−2 )2+(6−1 )2=√26=5.1 , d (BC )=√(2)2+ (5 )2=√29=5.4

d ( A' B' )=9 , d ( A ' C ' )=√32+152=√234=15.3 , d ( B' C ' )=√62+152=√261=16.2

k=93=15.3

5.1=16.2

5.4=3

From this, we also see that all the sides of ∆ A ' B ' C 'are three times larger than ∆ ABC.


Unit 1.8

GoThe origin is the center of dilation. Draw the dilation of each figure, given the scale factor.1. A (2,4 ) , B (−3,5 ) ,C (−1,2 );k=2

2. A (5 ,−5 ) , B (−4 ,−4 ) ,C (0,5 ) ;k=12


Unit 1.9

1.9 PARTITIONING A LINE SEGMENTStudent Notes


Unit 1.9

The Divided LineA Solidifying Understanding Task 9

Name_____________________ Hour___________

Graph the points (-7, -7) and (5, 9).1. How would you find the slope between these

points?

2. How could you use similar triangles to show that the line has a constant slope?

3. Review from previous math classes: match the forms for equations of lines, the name and the example (draw lines between the columns).

a. Ax + By = C x. Slope-Intercept Form 1. y = 3x + 9b. y = mx + b y. Standard Form 2. y – 3 = 3(x + 2)c. y – y1 = m(x – x1) z. Point-Slope Form 3. 3x – y = -9

4. a. These two lines are parallel: y = 4x + 3, y = 4x – 6Explain how you can tell this by looking at the equation and what does that mean?

b. These two lines are perpendicular: y = -2x + 3, y = ½ x – 6Explain how you can tell this by looking at the equation and what does that mean?


Unit 1.9

5. In your own words, explain how you would find out the missing information:a. If you graphed 3 non-colinear points and they made a triangle, how could you tell if that

triangle is an isosceles, equilateral or a right triangle?

b. If you graphed 4 non-colinear points, how could you determine if they formed a square, a rectangle, a rhombus or a parallelogram?

c. If you were given an equation of a line, and a point not on the line, how would you find an equation of a line through the point that is parallel to the given line?

d. If you were given an equation of a line, and a point not on the line, how would you find an equation of a line through the point that is perpendicular to the given line?

e. Given the equations of two non-parallel lines, how would you find their point of intersection (hint: systems)?

6. Using the points from #1, how could you find the middle of the line they create (the midpoint)?

7. Try to use what you found in #6 to help you determine a way to find a point on that line that partition or divides it into another fractional part. Use this to help you find a point of the line from #1 that is ¼ of the way up the line.


Unit 1.9

Ready, Set, Go!Name_______________________________ Hour____________Ready

1. Show two different ways to find the slope between the points (-1, 6) and (3, -2).

2. Write the equations of the points from #1 in standard form, slope-intercept form, and point-slope form. Compare and contrast the three forms.

3. a. Write the equation of the line that is parallel to the line from #1 and passes through the point (2, 3).

b. Write the equation of the line that is perpendicular to the line from #1 and passes through the point (2, 3).

4. What is the midpoint of the line formed in #1?

Set5. Subdivide the segment from #1 into fur equal parts. Identify the points that partition the

segment.

6. A segment with endpoints A(2, 4) and B(9, -2) is partitioned by a point C such that AC and CB form a 3:5 ratio. Find C.

7. Given segment PR with a point Q in between such that PQ and QR form a 2:7 ratio. Find R for P(-2, -5) and Q(2, 9).


Unit 1.9

Go8. Terhau County must choose a site for their new hospital. Most residents of the county live in

Givensville and would like the facility nearby. However, a small portion of residents live in the town of Pilette. Naturally, they would like easy access to the hospital, as well. As a compromise, the county committee has chosen a site along the road connecting the two towns, which reflects the population of each. The hospital site will partition the connecting road into the ratio 1: 2. Where is the location of the hospital?

9. The cross country team is planning a relay race from Hyrum to Smithfield. Local police have committed to provide a straight shot free of all other traffic. If each team consists of five runners, where should they position themselves in order to have each runner compete the same distance?


Hyrum

Smithfield

Unit 1

HOMEWORK HELP FOR STUDENTS AND PARENTS

http://www.khanacademy.org/math/geometry/triangles/v/similar-triangle-basics p http://euler.slu.edu/escher/index.php/Similarity_Transformations http://www.khanacademy.org/math/geometry/triangles/v/similarity-postulates http://www.mathwarehouse.com/geometry/similar/triangles/similar-triangle-theorems.php http://www.khanacademy.org/math/geometry/triangles/v/similar-triangle-example-

problems http://www.mathwarehouse.com/geometry/similar/triangles/side-splitter-theorem.php http://sketchexchange.keypress.com/sketch/view/388/6-3-use-similar-polygons http://www.khanacademy.org/math/geometry/triangles/v/congruent-and-similar-triangles http://staff.argyll.epsb.ca/jreed/math9/strand3/triangle_congruent.htm http://nlvm.usu.edu -National Library of Virtual Manipulatives: Transformations –

Dilation http://www.mathwarehouse.com/transformations/dilations/dilations-in-math.php http://www.khanacademy.org/math/geometry/triangles/v/similar-triangles http://nlvm.usu.edu/en/nav/frames_asid_296_g_4_t_3.html?

open=activities&from=search.html?qt=dilations http://math2.org/math/algebra/conics.htm http://www.khanacademy.org/math/geometry/triangles/v/pythagorean-theorem-proof-using-

similarity Fun Extension with transformations: Creating a Kaleidoscope with Geometer's

Sketchpad, www.youtube.com Math Help: http://www.webmath.com/ http://www.mathgoodies.com/students.html Math Practice: http://interactmath.com/ChapterContents.aspx http://www.maththatcounts.com/page6.html http://www.ies.co.jp/math/java/geo/similar.html


http://www.ies.co.jp/math/java/geo/similar.html

http://www.maththatcounts.com/page6.html

http://interactmath.com/ChapterContents.aspx

http://www.mathgoodies.com/students.html

http://www.webmath.com/

http://www.youtube.com/

http://www.khanacademy.org/math/geometry/triangles/v/pythagorean-theorem-proof-using-similarity

http://www.khanacademy.org/math/geometry/triangles/v/pythagorean-theorem-proof-using-similarity

http://math2.org/math/algebra/conics.htm

http://nlvm.usu.edu/en/nav/frames_asid_296_g_4_t_3.html?open=activities&from=search.html?qt=dilations

http://nlvm.usu.edu/en/nav/frames_asid_296_g_4_t_3.html?open=activities&from=search.html?qt=dilations

http://www.khanacademy.org/math/geometry/triangles/v/similar-triangles

http://www.mathwarehouse.com/transformations/dilations/dilations-in-math.php

http://nlvm.usu.edu/

http://staff.argyll.epsb.ca/jreed/math9/strand3/triangle_congruent.htm

http://www.khanacademy.org/math/geometry/triangles/v/congruent-and-similar-triangles

http://sketchexchange.keypress.com/sketch/view/388/6-3-use-similar-polygons

http://www.mathwarehouse.com/geometry/similar/triangles/side-splitter-theorem.php

http://www.khanacademy.org/math/geometry/triangles/v/similar-triangle-example-problems

http://www.khanacademy.org/math/geometry/triangles/v/similar-triangle-example-problems

http://www.mathwarehouse.com/geometry/similar/triangles/similar-triangle-theorems.php

http://www.khanacademy.org/math/geometry/triangles/v/similarity-postulates

http://euler.slu.edu/escher/index.php/Similarity_Transformations

http://www.khanacademy.org/math/geometry/triangles/v/similar-triangle-basics

Unit 1

SIMILARITY ASSESSMENTName_____________________ Hour___________

Short Answer Section

1. Determine whether each pair of ratios can form a proportion.

a) 8 :7 , 56 :49 b) 10 :15 , 8:12

2. Use each set of numbers to form two proportions.

a) 1 ,5 ,6 ,30 b) 4 , 6 ,12 , 18

3. The sides of a triangle measure 4 ,9 ,∧11. If the shortest side of a similar triangle measure 12, find the measure of the remaining sides of this similar triangle.

4. Determine whether each pair of triangles is similar. Explain why.

a) b)

Math Composer 1.1. 5http:/ / www.mathcomposer. com

59º

Math Composer 1.1. 5http: / /www. mathcomposer. com

37ºR

Q

M

L

P

5. A tower casts a shadow of 64 feet. A 6-foot pole near the tower casts a shadow of 8 feet long. How tall is the tower?

1. If HI = 10 and IJ = 10 and HL = 8, what is KL=_______?

2. What is the abbreviation of the theorem that states if all three sides of a triangle are proportionate to all three sides of another triangle, then the triangles are similar?


Unit 1

3. Create a dilation of segment AB through C with a scale factor of 2 to create segment EF. Find the lengths of EF, AC, BC, CE, and CF in cm.

9. Locate the center of dilation and scale factor in the following pair of triangles.

10. A segment with endpoints A(3,2) and B(6,11) is partitioned by a point C such that AC and CB form a 2:1 ratio. Find C.

Multiple-Choice Section (Circle your answer)

11. Pentagon VWXYZ is similar to pentagon JKLMN. What is the measurement of angle L?Math Composer 1.1.5http: / /www.mathcomposer.com

60º

150º?

Z

Y

X

W

V

N

M

L

K

J

a) 30 ° c) 150 °

b) 60 ° d) 120 °


A

B

C

Unit 1

12. Triangle PQR is similar to triangle DEF as shown. (not drawn to scale) Which describes the relationship between the corresponding sides of the two triangles?

Math Composer 1. 1. 5http: / /www. mathcomposer. com

9 cm

6 cm

6 cm

4 cm

F

E

DR

Q

P

a) PQDE

=46

b) PQDE

=64

c) PQEF

=49

d) PRDE

=66

13. Mr. Smith is having some photos enlarged for his studio. He wants to enlarge a photo that is 5 inches by 7 inches so the dimensions are 3 times larger than the original. How many times larger than the original photo will the area of the new photo be?

a) 3 b) 6 c) 9 d) 30

14. Ryan and Kathy each drew a triangle with an angle of 20 degrees. Under which condition would the triangles be similar?

a) if both are right triangles c) if the triangles have the same area

b) if both are obtuse triangles d) if the triangles have the same perimeter

15. For the figure at the right, if ABDB

=CBEB then what is true about lines

ED and CA?

a) they are congruent c) they are similar

b) they are parallel d) they are perpendicular

16. A point B(4,2) on a segment with endpoints A(2, -1) and C(x,y) partitions the segment in a 1:3 ratio. What is x?

a) -2 c) 0

b) 2 d) 4


Secondary II - PBworksnucc.pbworks.com/w/file/fetch/62335003/Unit_01_Similarity... · Web view1. 15...

Documents

Transcript of Secondary II - PBworksnucc.pbworks.com/w/file/fetch/62335003/Unit_01_Similarity... · Web view1. 15...